RESPONSE: Thank you for a great question. I have provided a visual aid (the picture at left) to reinforce the criticism (which you touched on) often raised about the apparent randomness of Christians picking and choosing which Old Testament laws apply and which ones don't. At any rate, my answer may surprise you. In a word, no. The Old Testament laws do not apply to us today (but not for the reasons given in the ridiculous picture).* The laws of the Old Testament were given to national Israel. They were designed to govern national Israel. We are not national Israel. Therefore, they do not apply to us today. What if I asked you, "Do the laws of Tennessee apply to the people in California?" The correct answer is no. The laws of Tennessee are designed to govern the state of Tennessee, not California, and vice versa. However, you might hesitate to say no because you immediately recognize that many laws in Tennessee are also laws in California. That is exactly the train of thought we need to follow in order to parse out the second half of your question. You said "it seems like some Christians pick and choose which ones they want to enforce and then just ignore the others." So, let me ask you another question: why do some people in Tennessee follow the same laws as the people in California? The answer is simple: many of the laws of Tennessee reflect universal moral principles that ought to be enforced everywhere. For instance, murder is a crime in both states. But that doesn't mean that the laws of Tennessee apply to the people in California. It simply means that there are universal moral obligations that apply to people wherever they are. Someone in Tennessee doesn't go to jail for murder because of the law against murder in California, but because murder is wrong, and the laws of Tennessee reflect that. Even so, many of the laws in Tennessee are different from California. California likely has different laws concerning alcohol sales and ownership of guns than Tennessee. So why the picking and choosing--enforcing some and just ignoring others? It is not an arbitrary decision--as if someone were just cherry picking the ones they liked. Again, it is because some of them reflect universal moral principles that ought to be enforced everywhere.
The same concept can be applied to the Old and New Testaments--to national Israel and the rest of us. Technically speaking, the law against murder that was given to the people of national Israel, for instance, does not apply to us today. No more than a law against Sunday alcohol sales in Tennessee would make it illegal to buy beer on the first day of the week in California. Now, before you flip your wig, please don't misunderstand me. That does not mean that it is okay to murder. The law given to Israel reflects a universal moral principle. Murder is wrong.
So the question becomes: how do we determine what laws reflect universal moral principles, and ought to be followed today? Well, if you were moving from California to Tennessee, one way to determine which laws reflect universal moral principles would be to find out which laws they have in common. Just compare them. In the same way, we can see what commands are repeated in the New Testament. If a command from the Old Testament is repeated in the New, it almost certainly reflects a transcendent moral principle.
Another approach is to look at teleology. The Greek word telos refers to the ultimate or intended end of a thing. Thus, teleology is the study of purpose. Are any of the laws of the Old Testament based upon the designed purpose of things? Sure. We don't need Old Testament prohibitions against bestiality to be repeated in the New Testament to know that it is wrong. It is a gross violation of God's design for human and animal sexuality. Any such laws--those based on the telos of a thing--would certainly apply to us today. Again, not because they were given to national Israel, but because they uphold the intended purpose of a thing.
I hope that you will forgive me for not going into specific laws here--does X apply, and why not Y? In this space I simply wanted to respond to your question (and the criticism raised by the meme) about the apparent randomness of picking some laws to apply and ignoring others. I hope that helps!
*I should clarify that when I say "does not apply to us," I do not mean, "is worthless to us." I mean "apply" in the same sense as in the analogy I used. In the way that the laws of the state of Tennessee do not apply to the people of California. Can the people of each state learn from the laws of the other? Certainly, and they often do. And in the case of the Old Testament, the law is given by God himself to Israel. So it is ALL good, ALL useful, and ALL important to understand. God's law reveals a tremendous amount to us about his nature. I don't want anyone to think that I am advocating any sort of neo-Marcionism (Marcion was an ancient heretic that believed in Jesus but rejected the Old Testament).